Turning Theorems into Plays

After overhearing the gossip of the house staff, the young Thomasina Coverly interrupts her algebra lesson on Fermat's Last Theorem to ask her tutor, Septimus Hodge, the meaning of "carnal embrace." Septimus's first explanation, "Throwing one's arms around a side of beef," proves unacceptable as Thomasina points out that it was a certain Mrs. Chater who was discovered in carnal embrace in the graden gazebo. "I don't think you have been candid with me Septimus," Thomasina insists. "A gazebo is not, after all, a meat larder." "Ah yes, I am ashamed," Septimus finally concedes. With blunt, almost medical precision, he then provides his talented pupil with a terse description of intercourse, explaining that carnal embrace is actually sexual congress between males and females...

Septimus: ... for purposes of procreation and pleasure. Fermat's last theorem, by contrast, asserts that when x, y, and z are whole numbers each raised to the power of n, the sum of the first two can never equal the third when n is greater than 2.

Thomasina: Eurghhh!

Septimus: Nevertheless, that is the theorem.

Cycloidal Areas without Calculus

For centuries mathematicians have been interested in curves that can be constructed by simple mechanical instruments. Among these curves are various cycloids used by Apollonius around 200 B.C. and by Ptolemy around 200 A.D. to describe the apparent motions of planets. The simplest cycloid is the curve traced out by a point on the circumference of a circular disk that rolls without slipping along a horizontal line; it forms a sequence of arches resting on the line.

WordWise: Arabic From A (Algebra) to Z (Zero)

Imagine that you were imprisoned on Alcatraz, and had just completed the fourth year of a 25-year sentence for trying to divide by zero. The cruel prison guards have tossed you in solitary confinement, and you've been stripped of your calculator! Okay, how do you figure out the number of days left in your sentence?

Chance Encounters: Probability: From Monte Carlo to Geometry

The interplay between probability and other areas of mathematics often occurs in unexpected places. Below you will see how probability can be applied to problems in calculus, the estimation of e, and even to geometrical proofs.

Kelvin's 100-year-old Bubble Conjecture Burst

The year 1994 brought striking news of the disproof of Lord Kelvin's 100-year-old conjecture by Denis Weaire and Robert Phelan of Trinity College, Dublin. To understand the problem, imagine an infinite soap bubble cluster filling all of space. Suppose that each bubble encloses exactly one cubic foot of air. What would be the most efficient shape for such a cluster, minimizing the total area or surface energy of all the interfaces between regions?

A Graduate School Primer

Who should go to graduate school? Where and why should they go? What is graduate school? This note is a biased answer to some of these questions. Keep in mind that the answers are provided by a person who believes that education in general is a wonderful thing, has no regrets about going to graduate school, and feels it led to an employment position that almost ideally suits his ambitions and temperament. This article addresses some of the preparations that should be taken by students planning to go to graduate school, but provides only the basic details. I do not pretend that it is a substitute for personal advice, which I strongly suggest you obtain before making such an important decision.

A Half-Dozen Mathematical Activities to Try with Friends

For the past several years I have run, at St. Mary's College of Maryland, a mathematics club with a unique flair. Although it has a social aspect, that is not its primary aim, nor is it focused on preparation for mathematics competitions. Rather, it attempts, through hands-on projects and discussions, to illustrate a sense of fascination and delight with mathematics, to foster a sense of curiosity, and to make sophisticated mathematics accessible, intriguing and fun. Students who participate range from freshmen to seniors, math majors to English majors. Even faculty members have enjoyed participating!

Problem Section:

S-31.

Proposed by Suat Nomli, Student, Bilkent University, Turkey. Construct an equiangular hexagon whose sides are 1, 2, 3, 4, 5, and 6.

S-32.

Proposed by William Bechem, Eastern Middle School and R.A. Rosenbaum, Wesleyan University. Prove that the incenter of a triangle lies on its Euler line (the line through the circumcenter and the centroid), if and only if the triangle is isosceles.

S-33.

Proposed by Mircea Ghita, Stuyvesant High School. Solve for real x:
(4^x)(9^{1/x}) + (4^{1/x})(9^x)=72.

S-34.

Proposed by E. M. Kaye, Vancouver, BC. Can the sum of the squares of 61 consecutive integers ever be a perfect square?

The Final Exam: Math Horizons t-shirt Contest

(To read the winning entry to April's Final Exam, click here.)

We all agree that math is beautiful. If you wander around a math department you will undoubtedly see a professor or student who thinks math is so beautiful that she is wearing some on her t-shirt. Over the years we've seen lots of math t-shirts: some were clever, others profound, some beautiful, most were just silly. Visual jokes are popular, and we've all seen many Penrose and Escher graphics and Sidney Harris cartoons. The American Statistical Association sells a shirt bearing the phrase "I am statistically significant." top ten lists are a perennial fashion statement among the numerical cognoscenti: Top Ten Proof Techniques -- 1. Intimidation 2. Gesticulation 3. Circular Reasoning (See #3.) 4. Denial ...; Top Ten Mathematical Pick-up lines -- 1. Voulez-vous Cauchy avec moi ce soir? 2. Hey baby, what's your sine? 3. Your LaPlace or mine? 4. Have we met? You Riemann me of someone. Our all-time favorite may just be the emphatic CARPE THEOREM.

By now you've guessed it: the contest is to design an official Math Horizons t-shirt. The winning design will embody the wit, wonder, and playfulness that the magazine strives for in its articles. We will also award prizes for best departmental t-shirt, best pun, and best graphic. Send your design ideas to skennedy@carleton.edu, or, if you want to try to influence the judges by sending us an actual t-shirt (XXL) mail to Steve Kennedy, Math Department, Carleton College, Northfield, MN 55057. The creators of winning designs will receive, duh, a Math Horizons t-shirt and, of course, the usual everlasting fame and glory. Deadlines for entries is November 1, 1999.